An own-goal by Craig Wedderburn was sandwiched between strikes from David Healy and Nikica Jelavic before Kerkar sealed the win late on.
Rangers had crashed to defeat to Falkirk in the League Cup earlier in the season but ensured there would be no shocks against their Irn-Bru Second Division opponents on Sunday.
Rangers were boosted by the return of top goalscorer Jelavic from injury for the cup clash.
John Fleck also started at Gayfield, with Kyle Lafferty injured and Steven Davis suspended following Monday’s Scottish Premier League win over Motherwell.
Arbroath were unchanged from their defeat against Dumbarton, meaning player-boss Paul Sheerin started for the home side.
Rangers tried to pile the pressure on straight away and Healy tried his luck with a shot that fell tamely over the crossbar.
Arbroath had a decent chance when skipper for the day Carlos Bocanegra slipped, allowing Steven Doris to burst through on goal only to drag his shot wide.
Rangers goalkeeper Allan McGregor was then called into action to produce an impressive save to tip a well-struck shot from Josh Falkingham just over.
However, it was Rangers who opened the scoring after 17 minutes when Maurice Edu’s pass was collected by Jelavic and he slipped through to Healy who fired past goalkeeper Darren Hill.
Hopes of an Arbroath comeback were dealt a blow when Rangers doubled their lead after 22 minutes.
Jelavic exchanged a one-two with Edu before whipping a dangerous ball into the box that was turned into his own net by Wedderburn.
Rangers were threatening again when Fleck released Lee Wallace down the left flank and he squared for Jelavic who blasted wide of target.
Hill then came to the rescue with a good save to block a powerful effort from Sone Aluko after Edu teed up the shot.
The tie could have been over as a contest within a minute of the restart had Fleck been on target but he sent his diving header past the upright.
At the other end, Lee Sibanda claimed he was fouled in the box by Dorin Goian but there was no penalty award for the home side from referee Calum Murray.
Instead, it was Rangers who had the net bulging again after 59 minutes when Aluko threaded into the path of Jelavic and the striker’s effort smashed off the inside of the post and over the line.
There was concern for Rangers when Kyle Bartley collapsed to the turf clutching his leg and, although he appeared to be fine after receiving treatment, he was swiftly replaced by Jordan McMillan.
Kerkar had already been thrown into action for Wallace and he added his name to the scoresheet after 77 minutes.
Sasa Papac picked out Healy at the back post and he swept into the path of Kerkar to rifle high into the net and put the gloss on the win.